Finite Group Databasebeta

← #29650079

Subgroups — #29650079

Order 15616.

648 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455862C211yesmaximal, characteristic
2#21454968C211yesmaximal, characteristic, complemented
3#21454842C211yesmaximal, characteristic, complemented
4#21444686C211yesmaximal, characteristic, complemented
5#21444679C211yesmaximal, characteristic, complemented
6#21444658C211yesmaximal, characteristic, complemented
7#21444651C211yesmaximal, characteristic, complemented
8#21444640C211yesmaximal, characteristic, complemented
9#21444634C211yesmaximal, characteristic, complemented
10#21444616C211yesmaximal, characteristic, complemented
11#21444611C211yesmaximal, characteristic, complemented
12#21444607C211yesmaximal, characteristic, complemented
13#21444245C211yesmaximal, characteristic, complemented
14#21444119C211yesmaximal, characteristic, complemented
15C61×#1482C211yesmaximal, characteristic, complemented
16C2×C4×D244C2211yescharacteristic
17#16840095C411yescharacteristic
18#16840078C411yescharacteristic
19#16840059C411yescharacteristic
20#16840059C411yescharacteristic
21#16839974C411yescharacteristic
22#16839974C411yescharacteristic
23#16839912C411yescharacteristic
24#16839895C411yescharacteristic
25#16839893C2211yescharacteristic, complemented
26Q8×#92418C2211yescharacteristic, complemented
27C4×#11758201C2211yescharacteristic
28C4×#11758201C2211yescharacteristic
29C4×#11758199C2211yescharacteristic
30C4×#11758199C2211yescharacteristic
31#16839840—21no—
32#16839840C2211yescharacteristic, complemented
33#16839834—21no—
34C8×D244—21no—
35C8×D244—21no—
36D4×#92418C2211yescharacteristic, complemented
37#16839827C2211yescharacteristic
38#16839827C2211yescharacteristic
39#16839827C2211yescharacteristic
40#16839827C2211yescharacteristic
41#16839823—21no—
42#16839815C2211yescharacteristic
43#16839815C2211yescharacteristic
44#16839789—21no—
45#16839786C2211yescharacteristic
46C22⋊C8×D61—21no—
47C22⋊C8×D61—21no—
48#16839514C2211yescharacteristic, complemented
49#16839514C2211yescharacteristic, complemented
50#16839512C2211yescharacteristic, complemented
51#16839512C2211yescharacteristic, complemented
52#16839491C2211yescharacteristic
53#16839491C2211yescharacteristic
54#16839491C2211yescharacteristic
55#16839491C2211yescharacteristic
56#16839484C2211yescharacteristic, complemented
57#16839484C2211yescharacteristic, complemented
58#16839483C2211yescharacteristic, complemented
59C8×#92418C2211yescharacteristic, complemented
60C61×#529C2211yescharacteristic
61C61×#431C2211yescharacteristic
62C61×#431C2211yescharacteristic
63C61×Q8⋊C8C2211yescharacteristic, complemented
64C61×Q8⋊C8C2211yescharacteristic, complemented
65C61×D4⋊C8C2211yescharacteristic, complemented
66C61×D4⋊C8C2211yescharacteristic, complemented
67C22×D244C2×C411yescharacteristic
68C22×C4×D61—21no—
69C2×#11758201C2×C411yescharacteristic
70C2×#11758201C2×C411yescharacteristic
71C2×#11758199C2×C411yescharacteristic
72C2×#11758199C2×C411yescharacteristic
73C2×#11758194—21no—
74C2×#11758194—21no—
75C22×#92418—21no—
76C2×C8×D61—21no—
77C2×C8×D61—21no—
78C2×C8×D61—21no—
79C2×C8×D61—21no—
80C4×D244—21no—
81C4×D244—21no—
82C4×D244C2×C411yescharacteristic
83C4×D244C2×C411yescharacteristic
84C4×D244C2311yescharacteristic
85C4×D244C2311yescharacteristic
86C2×C4⋊Dic61C2×C411yescharacteristic
87C2×#11758198—21no—
88#14519710C2×C411yescharacteristic
89#14519710C2×C411yescharacteristic
90#14519708C2×C411yescharacteristic
91#14519708C2×C411yescharacteristic
92C4⋊#92418—21no—
93C4⋊#92418—21no—
94C4⋊#92418—21no—
95C4⋊#92418C2×C411yescharacteristic
96C4⋊#92418C2×C411yescharacteristic
97C4⋊#92418D411yescharacteristic
98C4⋊#92418D411yescharacteristic
99C4⋊#92418C2311yescharacteristic
100C4⋊#92418C2311yescharacteristic
101#14519701C2×C411yescharacteristic
102#14519701C2×C411yescharacteristic
103#14519698C2×C411yescharacteristic
104#14519698C2×C411yescharacteristic
105#14519697C2×C411yescharacteristic
106#14519697C2×C411yescharacteristic
107#14519697C2×C411yescharacteristic
108#14519697C2×C411yescharacteristic
109C22⋊#92418—21no—
110C22⋊#92418—21no—
111C22⋊#92418C2×C411yescharacteristic
112C22⋊#92418C2×C411yescharacteristic
113C22⋊#92418D411yescharacteristic
114C22⋊#92418D411yescharacteristic
115#14519689—21no—
116#14519689—21no—
117#14519689—21no—
118#14519689—21no—
119#14519688—21no—
120C4×#92418—21no—
121C4×#92418C2311yescharacteristic
122C4×#92418C2311yescharacteristic
123C4×#92418C2311yescharacteristic
124C4×#92418C2311yescharacteristic
125#126×C61—21no—
126#125×C61C2×C411yescharacteristic
127#124×C61C2×C411yescharacteristic
128#123×C61—21no—
129C22⋊Q8×C61C2×C411yescharacteristic
130#121×C61C2×C411yescharacteristic
131C4×Q8×C61C2311yescharacteristic
132C4×D4×C61—21no—
133C4×D4×C61C2311yescharacteristic
134#117×C61—21no—
135C2×C42×C61C2311yescharacteristic
136C4⋊C8×C61—21no—
137C4⋊C8×C61C2311yescharacteristic
138C4⋊C8×C61C2311yescharacteristic
139C22⋊C8×C61—21no—
140C22⋊C8×C61—21no—
141C4×C8×C61—21no—
142C4×C8×C61C2311yescharacteristic
143C4×C8×C61C2311yescharacteristic
144C23×D61—21no—
145C2×Q8×C61C22×C411yescharacteristic
146C2×D4×C61—21no—
147C2×D4×C61C22×C411yescharacteristic
148C22×C4×C61—21no—
149C22×C4×C61C22×C411yescharacteristic
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61#4111yescharacteristic
152C22×Dic61—21no—
153C2×D244—21no—
154C2×D244—22no—
155C2×D244—22no—
156C2×D244C22×C411yescharacteristic
157C2×D244C22×C411yescharacteristic
158C2×C4×D61—21no—
159C2×C4×D61—21no—
160C2×C4×D61—21no—
161C2×C4×D61—21no—
162C2×C4×D61—81no—
163C2×C4×D61—81no—
164C2×C8×C61—21no—
165C2×C8×C61—21no—
166C2×C8×C61—21no—
167C2×C8×C61—21no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61C22×C411yescharacteristic
173C4⋊C4×C61C22×C411yescharacteristic
174C22⋊C4×C61—21no—
175C22⋊C4×C61—21no—
176C22⋊C4×C61—21no—
177C22⋊C4×C61—21no—
178C22⋊C4×C61—21no—
179C42×C61—21no—
180C42×C61C22×C411yescharacteristic
181C42×C61C2×D411yescharacteristic
182C42×C61#4111yescharacteristic
183C42×C61C2411yescharacteristic
184#11758201—22no—
185#11758201—22no—
186#11758199—22no—
187#11758199—22no—
188#11758198—21no—
189#11758198—21no—
190#11758198—21no—
191#11758198—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61C22×C411yescharacteristic
194C4⋊Dic61C22×C411yescharacteristic
195#11758194—81no—
196#11758194—81no—
197C2×#92418—21no—
198C2×#92418—21no—
199C2×#92418—81no—
200C2×#92418C22×C411yescharacteristic
201C2×#92418C22×C411yescharacteristic
202C2×#92418C22×C411yescharacteristic
203C2×#92418C22×C411yescharacteristic
204C2×#92418C2×D411yescharacteristic
205C2×#92418C2×D411yescharacteristic
206C2×#92418C2×D411yescharacteristic
207C2×#92418C2×D411yescharacteristic
208C8×D61—81no—
209C8×D61—81no—
210C8×D61—81no—
211C8×D61—81no—
212C23×C61—21no—
213C23×C61C4×D411yescharacteristic
214C22×D61—21no—
215C22×D61—21no—
216C22×D61—24no—
217C22×D61—81no—
218C22×D61—81no—
219Q8×C61—21no—
220Q8×C61—21no—
221D4×C61—21no—
222D4×C61—21no—
223D4×C61—81no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—41no—
232C2×C4×C61—41no—
233C2×C4×C61—41no—
234C2×C4×C61C4×D411yescharacteristic
235C2×C4×C61C4×D411yescharacteristic
236C2×C4×C61C4×D411yescharacteristic
237C2×C4×C61C23×C411yescharacteristic
238C2×C4×C61C22×D411yescharacteristic
239C2×C4×C61C2×#4111yescharacteristic
240C2×Dic61—21no—
241C2×Dic61—21no—
242C2×Dic61—81no—
243D244—24no—
244D244—21no—
245D244—21no—
246D244#13112yes—
247D244#13112yes—
248C4×D61—41no—
249C4×D61—41no—
250C4×D61—41no—
251C4×D61—41no—
252C4×D61—81no—
253C4×D61—81no—
254C8×C61—41no—
255C8×C61—41no—
256C8×C61—41no—
257C8×C61—41no—
258#92418—21no—
259#92418—21no—
260#92418—21no—
261#92418—21no—
262#92418—21no—
263#92418—21no—
264#92418—21no—
265#92418—21no—
266#92418—41no—
267#92418—41no—
268#17918—611nomaximal, Sylow
269C22×C61—21no—
270C22×C61—21no—
271C22×C61—21no—
272C22×C61—41no—
273C22×C61—41no—
274C22×C61—41no—
275C22×C61C2×C4×D411yescharacteristic
276C2×D61—22no—
277C2×D61—22no—
278C2×D61—41no—
279C2×D61—41no—
280C2×D61—41no—
281C2×D61—41no—
282C2×D61—84no—
283C4×C61—21no—
284C4×C61—21no—
285C4×C61—21no—
286C4×C61—21no—
287C4×C61—41no—
288C4×C61—41no—
289C4×C61—41no—
290C4×C61C2×#13111yescharacteristic
291C4×C61C2×#13111yescharacteristic
292C4×C61#57611yescharacteristic
293C4×C61#57711yescharacteristic
294Dic61—41no—
295Dic61—41no—
296#3105—611no—
297#2993—611no—
298#2972—611no—
299#1586—611no—
300#1585—611no—
301#1582—611no—
302#1581—611no—
303#1580—611no—
304SD16⋊C8—611no—
305C8×SD16—611no—
306C8×D8—611no—
307#1574—611no—
308#1486—611no—
309#1482D6111yescharacteristic, complemented
310#1454—611no—
311C2×C61—41no—
312C2×C61—81no—
313C2×C61#292911yescharacteristic
314C2×C61#294611yescharacteristic
315C2×C61#295711yescharacteristic
316D61—42no—
317D61—42no—
318#529C2×D6111yescharacteristic
319C2×C4×D4—611no—
320#484—611no—
321#481—611no—
322#478—611no—
323#478—611no—
324#463—611no—
325#463—611no—
326#450—611no—
327#447—611no—
328#446—611no—
329C8×Q8—611no—
330C4×SD16—611no—
331C4×SD16—611no—
332C4×D8—611no—
333C4×D8—611no—
334#436—611no—
335#436—1221no—
336#435—1221no—
337C8×D4—611no—
338C8×D4—611no—
339C8×D4—611no—
340C8×D4—611no—
341C8×D4—611no—
342C8×D4—1221no—
343C8×D4—1221no—
344#433—1221no—
345#431—611no—
346#431—611no—
347#431C2×D6111yescharacteristic
348#431C2×D6111yescharacteristic
349#424—1221no—
350#423—611no—
351C2×C22⋊C8—1221no—
352C2×C22⋊C8—1221no—
353#335—611no—
354#335—611no—
355#334—611no—
356#334—611no—
357Q8⋊C8C2×D6111yescharacteristic, complemented
358Q8⋊C8C2×D6111yescharacteristic, complemented
359D4⋊C8—611no—
360D4⋊C8—611no—
361D4⋊C8—611no—
362D4⋊C8—611no—
363D4⋊C8C2×D6111yescharacteristic, complemented
364D4⋊C8C2×D6111yescharacteristic, complemented
365#322—611no—
366#322—611no—
367#322—611no—
368C82—611no—
369C61#1791811yescharacteristic, complemented, Sylow
370C22×D4—611no—
371C23×C4—1221no—
372C2×SD16—611no—
373C2×SD16—611no—
374C2×D8—611no—
375C2×D8—611no—
376C2×M16—1221no—
377C2×M16—1221no—
378C22×C8—1221no—
379C22×C8—1221no—
380C22×C8—1221no—
381C22×C8—1221no—
382C22×C8—1221no—
383#126—21no—
384#125C4×D6111yescharacteristic
385#124C4×D6111yescharacteristic
386#123—21no—
387C22⋊Q8C4×D6111yescharacteristic
388#121C4×D6111yescharacteristic
389C4×Q8C22×D6111yescharacteristic
390C4×D4—21no—
391C4×D4—611no—
392C4×D4—611no—
393C4×D4—611no—
394C4×D4—611no—
395C4×D4—1221no—
396C4×D4—1221no—
397C4×D4C22×D6111yescharacteristic
398#117—21no—
399C2×C4⋊C4—611no—
400C2×C22⋊C4—1221no—
401C2×C42C22×D6111yescharacteristic
402#107—611no—
403#107—611no—
404#106—611no—
405#106—611no—
406C4⋊C8—21no—
407C4⋊C8—611no—
408C4⋊C8—611no—
409C4⋊C8—611no—
410C4⋊C8—611no—
411C4⋊C8—611no—
412C4⋊C8—611no—
413C4⋊C8—1221no—
414C4⋊C8—1221no—
415C4⋊C8—1221no—
416C4⋊C8C22×D6111yescharacteristic
417C4⋊C8C22×D6111yescharacteristic
418#103—611no—
419#103—611no—
420#102—611no—
421#102—611no—
422#102—611no—
423#102—611no—
424#102—611no—
425#102—611no—
426C22⋊C8—21no—
427C22⋊C8—21no—
428C22⋊C8—611no—
429C22⋊C8—611no—
430C22⋊C8—611no—
431C22⋊C8—611no—
432C22⋊C8—1221no—
433C22⋊C8—1221no—
434C22⋊C8—1221no—
435C22⋊C8—1221no—
436C22⋊C8—1221no—
437C22⋊C8—1221no—
438#97—1221no—
439C4×C8—21no—
440C4×C8—611no—
441C4×C8—611no—
442C4×C8—611no—
443C4×C8—611no—
444C4×C8—1221no—
445C4×C8C22×D6111yescharacteristic
446C4×C8C22×D6111yescharacteristic
447C24—1221no—
448C2×Q8C2×C4×D6111yescharacteristic
449C2×D4—21no—
450C2×D4—611no—
451C2×D4—611no—
452C2×D4—1222no—
453C2×D4—1222no—
454C2×D4—1221no—
455C2×D4C2×C4×D6111yescharacteristic
456C22×C4—21no—
457C22×C4—1221no—
458C22×C4—1221no—
459C22×C4—1221no—
460C22×C4—1221no—
461C22×C4—1221no—
462C22×C4—4881no—
463C22×C4—4881no—
464C22×C4C2×C4×D6111yescharacteristic
465C22×C4D4×D6111yescharacteristic
466C22×C4#1175822411yescharacteristic
467SD16—1222no—
468SD16—1222no—
469D8—1222no—
470D8—1222no—
471M16—4881no—
472M16—4881no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—611no—
478C2×C8—611no—
479C2×C8—611no—
480C2×C8—611no—
481C2×C8—611no—
482C2×C8—611no—
483C2×C8—611no—
484C2×C8—611no—
485C2×C8—1221no—
486C2×C8—1221no—
487C2×C8—4881no—
488C2×C8—4881no—
489C2×C8—4881no—
490C2×C8—4881no—
491C2×C8—4881no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—611no—
497C4⋊C4—611no—
498C4⋊C4—1221no—
499C4⋊C4C2×C4×D6111yescharacteristic
500C4⋊C4C2×C4×D6111yescharacteristic
501C22⋊C4—21no—
502C22⋊C4—21no—
503C22⋊C4—21no—
504C22⋊C4—21no—
505C22⋊C4—21no—
506C22⋊C4—1221no—
507C22⋊C4—1221no—
508C22⋊C4—1221no—
509C22⋊C4—1221no—
510C42—21no—
511C42C2×C4×D6111yescharacteristic
512C42D4×D6111yescharacteristic
513C42#1175822411yescharacteristic
514C42C23×D6111yescharacteristic
515C23—21no—
516C23—1224no—
517C23—1221no—
518C23—1221no—
519C23—4881no—
520C23—4881no—
521C23#1451973711yescharacteristic
522Q8—21no—
523Q8—21no—
524D4—21no—
525D4—21no—
526D4—81no—
527D4—612no—
528D4—612no—
529D4—1224no—
530D4—1221no—
531D4—1221no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—1221no—
543C2×C4—1221no—
544C2×C4—4881no—
545C2×C4—4881no—
546C2×C4—4881no—
547C2×C4—2441no—
548C2×C4—2441no—
549C2×C4—2441no—
550C2×C4—2441no—
551C2×C4#1451973711yescharacteristic
552C2×C4#1451973711yescharacteristic
553C2×C4#1451973711yescharacteristic
554C2×C4C22×C4×D6111yescharacteristic
555C2×C4C2×D4×D6111yescharacteristic
556C2×C4C2×#1175822411yescharacteristic
557C8—41no—
558C8—41no—
559C8—41no—
560C8—41no—
561C8—1221no—
562C8—1221no—
563C8—1221no—
564C8—1221no—
565C8—1221no—
566C8—1221no—
567C8—1221no—
568C8—1221no—
569C8—2441no—
570C8—2441no—
571C22—21no—
572C22—21no—
573C22—21no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—1222no—
578C22—1222no—
579C22—4884no—
580C22—2441no—
581C22—2441no—
582C22—2441no—
583C22—2441no—
584C22C2×#1451973711yescharacteristic
585C4—21no—
586C4—21no—
587C4—21no—
588C4—21no—
589C4—41no—
590C4—41no—
591C4—41no—
592C4—2441no—
593C4—2441no—
594C4#131×D6111yescharacteristic
595C4#131×D6111yescharacteristic
596C4#1684056411yescharacteristic
597C4#1684056911yescharacteristic
598C2—41no—
599C2—81no—
600C2—2442no—
601C2—2442no—
602C2#2145451611yescharacteristic
603C2#2145465411yescharacteristic
604C2#2145473211yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#1482C2
socleC22×C61C2×C4×D4
prime radical#16839895C4
cosocleC42×C61C24
hypercenter#1482D61
hypocenterC61#17918
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1482

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #21454968C2semidirect
3. #21454842C2semidirect
4. #21444686C2semidirect
5. #21444679C2semidirect
6. #21444658C2semidirect
7. #21444651C2semidirect
8. #21444640C2semidirect
9. #21444634C2semidirect
10. #21444616C2semidirect
11. #21444611C2semidirect
12. #21444607C2semidirect
13. #21444245C2semidirect
14. #21444119C2semidirect
15. C61×#1482C2semidirect
25. #16839893C22semidirect
26. Q8×#92418C22semidirect
32. #16839840C22semidirect
36. D4×#92418C22semidirect
48. #16839514C22semidirect
49. #16839514C22semidirect
50. #16839512C22semidirect
51. #16839512C22semidirect
56. #16839484C22semidirect
57. #16839484C22semidirect
58. #16839483C22semidirect
59. C8×#92418C22semidirect
63. C61×Q8⋊C8C22semidirect
64. C61×Q8⋊C8C22semidirect
65. C61×D4⋊C8C22semidirect
66. C61×D4⋊C8C22semidirect
309. #1482D61semidirect
357. Q8⋊C8C2×D61semidirect
358. Q8⋊C8C2×D61semidirect
363. D4⋊C8C2×D61semidirect
364. D4⋊C8C2×D61semidirect
369. C61#17918semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#214558622
#214549682
#214548422
#214446862
#214446792
#214446582
#214446512
#214446402
#214446342
#214446162
#214446112
#214446072
#214442452
#214441192
C61×#14822
#1791861